{ Start of file Joseph5.Pas ************************************************}
{$A+,B-,D+,E+,F-,I+,L+,N+,O-,R-,S+,V+}
{$M 16384,0,655360}
Program Joseph5; { Graphical Orbits in Josephus permutations }
uses Crt, Graph; { Turbo Pascal 6.0 interface }
const
Name = 'Joseph5 - Graphical Orbits in Josephus permutations';
Version = 'Version 1.10, last revised: 1994-04-23, 0600 hours';
Author = 'Copyright (c) 1981-1994 by author: Harry J. Smith,';
Address = '19628 Via Monte Dr., Saratoga CA 95070. All rights reserved.';
{***************************************************************************}
{ Challenge given in REC Jan/Feb/Mar 1991 on page 24.
"The Thousand Roman Slaves" suggested by Steve Wagler
Original Problem: 1000 slaves in a circle, numbered 1 to 1000, are all to
be shot except one lucky survivor. The order of shooting is 1, 3, 5, etc.,
always alternating, and always immediately removing the fallen bodies. Once
a body has fallen, it is no longer considered part of the circle for
purposes of future counting and alternation. Example, n=6: Shoot 1,3,5,2,6;
4 survives.
The General Problem: There is an ordered set of n objects arranged in a
circle with object i (1 <= i <= n) in position i. All n objects are
selected and removed in a certain order and placed in a new circle with
the new position number k beings the order of selection. Object f is
selected first. After each selection, m minus 1 of the remaining objects
following the one removed are skipped and the next object is then
selected. We are interested in the nature of the permutation generated by
this process, its fixed elements, and in particular the original position
L of the last object selected. Note that m and f can be as low as 1 and
can be larger than n.
See Knuth, The Art of Computer Programming, Vol. 1, Pages 158-159, 181,
516, 521 and Vol. 3, Pages 18-19, 579.
Examples: (See Joseph1.Exe)
--------------------
n = 1000, m = 2, f = 1: Object 976 was selected last!
There is 1 fixed element: 1.
--------------------
Note: When m = 2 and f = 1, the last object selected = 2^P - 2(2^P - n)
= 2n - 2^P, where 2^P is the next power of 2 greater than or equal to n.
In this case n = 1000 so the last object selected = 2000 - 1024 = 976.
}
{ Developed in Turbo Pascal 6.0 }
const
Maxn = 15000; { The maximum total number of objects in the circle }
type
ColorType = 0 .. MaxColors; { Turbo Pascal type for colors }
Reals = Double; { Change Double to Real if no coprocessor }
var
A : array[1..Maxn] of Integer; { Working array of objects }
{ Ends up with A[i] = when object i was selected }
B : array[1..Maxn] of Integer; { Record of objects selected }
{ Ends up with B[i] = object # selected on k'th selection }
Ch : Char; { Character input by ReadKey }
i : Integer; { Utility index }
n : Integer; { The total number of objects in the circle to start with }
r : Integer; { Remaining number of objects in the circle }
m : Integer; { m for selecting every m'th object }
k : Integer; { Number of objects selected (killed) so far }
f : Integer; { # of first object to be selected }
c : Integer; { Index to current object being scanned }
p : Integer; { Index to previous object }
s : Integer; { Number of fixed (stationary) elements in permutation }
Maxm : Integer; { Maximum m for selecting every m'th object }
{ Graphics variables }
x : Integer; { Current value of x }
y : Integer; { Next step on circle, x --> y }
AspectR : Reals; { Aspect ratio, EGA Hi = 10000 / 7750 = 1.2903225806 }
XAsp : Word; { Horizontal aspect }
YAsp : Word; { Vertical aspect }
Ra : Reals; { Radius of circle }
SlopeX : Reals; { Slope of transform of X to pixels }
SlopeY : Reals; { Slope of transform of Y to pixels }
X0 : Reals; { X of point in transform of X to pixels }
Xp0 : Reals; { Xp of point in transform of X to pixels }
Y0 : Reals; { Y of point in transform of Y to pixels }
Yp0 : Reals; { Yp of point in transform of Y to pixels }
Border : Reals; { Size of border on screen in X pixels }
Device : Integer; { Turbo Pascal graphics device code }
Mode : Integer; { Turbo Pascal graphics device mode code }
TH : Integer; { Text height = TextHeight('M') }
TH3 : Integer; { Text height + 3 }
MaxXPix : Word; { Max pixel number in X direction, [0, MaxXPix] }
MaxYPix : Word; { Max pixel number in Y direction, [0, MaxYPix] }
PreRead : Boolean; { Ch has been preread by PlotIt procedure }
Color : ColorType; { Turbo Pascal color code }
Palette : PaletteType; { Turbo Pascal color palette }
X1, X2, X3 : Reals; { Points to plot in real coordinates }
Y1, Y2, Y3 : Reals;
{--------------------------------------}
procedure ExitProc; { Exit the program }
begin
RestoreCrtMode;
CloseGraph;
Halt(0);
end; { ExitProc }
{--------------------------------------}
function I2St( I : LongInt) : String; { Convert Integer to a String }
var St : String[11];
begin
Str( I, St); I2St:= St;
end; { I2St }
{--------------------------------------}
function ForceColor( Color : ColorType) : ColorType;
begin
Color:= Color mod Palette.Size;
IF Color = 0 then Color:= 1;
SetColor( Color);
ForceColor:= Color;
end; { ForceColor }
{--------------------------------------}
procedure WriteIn( Ind : Integer; St : String); { Write a line indented }
begin
for i:= 1 to Ind do Write(' ');
WriteLn( St);
end; { WriteIn }
{--------------------------------------}
procedure WriteC( St : String); { Write a line centered }
begin
for i:= 1 to ((78 - Length( St)) div 2) do Write(' ');
WriteLn( St);
end; { WriteIn }
{--------------------------------------}
procedure TestIt; { Halt the run if ESC typed }
begin
while Keypressed do Ch:= ReadKey;
if Ord( Ch) = 27 then ExitProc;
end; { TestIt }
{--------------------------------------}
procedure Pause; { Pause and allow operator to escape }
begin
Write('Type any key to continue, or Esc to quit ... ');
if KeyPressed then TestIt; { Strip keyboard input }
Ch:= ReadKey;
if Ord( Ch) = 27 then ExitProc;
WriteLn; WriteLn;
end; { Pause }
{--------------------------------------}
procedure Story; { Tell the story of what the Josephus problem is }
begin
WriteIn( 4,
'The General Problem: There is an ordered set of n objects arranged in a'
); WriteIn( 4,
'circle with object i (1 <= i <= n) in position i. All n objects are'
); WriteIn( 4,
'selected and removed in a certain order and placed in a new circle with'
); WriteIn( 4,
'the new position number k beings the order of selection. Object f is'
); WriteIn( 4,
'selected first. After each selection, m minus 1 of the remaining objects'
); WriteIn( 4,
'following the one removed are skipped and the next object is then'
); WriteIn( 4,
'selected. We are interested in the nature of the permutation generated by'
); WriteIn( 4,
'this process, its fixed elements, and in particular the original position'
); WriteIn( 4,
'L of the last object selected. Note that m and f can be as low as 1 and'
); WriteIn( 4,
'can be larger than n.'
); WriteLn;
end; { Story }
{--------------------------------------}
procedure ReadInt( Mess : String; Min, Max, Nom : Integer;
var I : Integer); { Read in an integer from keyboard }
var
St : String[255];
Stat : Integer;
LI : LongInt;
begin
repeat
WriteLn( Mess);
Write(' [', Min, ', ', Max, '] (ENTER => ', Nom, '): ');
ReadLn( St);
Val( St, LI, Stat);
until ((Stat = 0) and (LI >= Min) and (LI <= Max)) or ( Length( St) = 0);
if Length( St) = 0 then LI:= Nom;
I:= LI;
WriteLn('Input = ', I);
WriteLn;
end; { ReadInt }
{--------------------------------------}
procedure InitHead; { Initialize header }
begin
TextBackground( Blue);
TextColor( Yellow);
ClrScr;
WriteLn;
WriteC( Name);
WriteC( Version);
WriteC( Author);
WriteC( Address);
WriteLn;
Story; { Tell the story of what the Josephus problem is }
WriteLn(
' This program uses the mod function to speed up the solution.');
WriteLn;
end; { InitHead }
{--------------------------------------}
procedure Init; { Initialize program }
begin
InitHead; { Init header before and after InitGraph, InitGraph may hang }
Device:= Detect; Mode:= 0;
InitGraph( Device, Mode, ''); { Determine type of graphics device }
while Device < 0 do begin
WriteLn('*** Cannot initialize graphics ***'); Pause;
Mode:= 0;
ReadInt('Input Device #, 0 for Automatic Detection', 0, 10, 0, Device);
if Device <> 0 then ReadInt('Input Mode #', 0, 5, 0, Mode);
InitGraph( Device, Mode, '');
end;
MaxXPix:= GetMaxX; MaxYPix:= GetMaxY; { Initialize graphics }
GetAspectRatio( XAsp, YAsp);
AspectR:= YAsp / XAsp;
GetPalette( Palette);
IF Palette.Size < 2 then Palette.Size:= 2;
TH:= TextHeight('M');
TH3:= TH + 3;
Border:= 4 * TH3 * AspectR;
RestoreCrtMode;
InitHead;
WriteLn('Device, Mode, MaxX, MaxY = ',
Device, ' ', Mode, ' ', MaxXPix, ' ', MaxYPix);
Write('Palette.Size: Colors = ', Palette.Size, ':');
FOR I:= 0 TO Palette.Size-1 DO
Write(' ', Palette.Colors[I]); { Show hardware information }
WriteLn;
WriteLn('Screen pixel aspect ratio = ', AspectR:0:9);
WriteLn;
Pause; { Pause and allow operator to escape }
{ Read in n, m, and f }
ReadInt('Input n, the total number of objects', 1, Maxn, 1000, n);
ReadInt('Input m, positions to move for each choice', 1, MaxInt, 2, m);
ReadInt('Input f, object number to select first', 1, n, 1, f);
SetGraphMode( Mode);
IF Palette.Size > 8 then SetBkColor( Blue);
Color:= ForceColor( Yellow);
PreRead:= False;
end; { Init }
{--------------------------------------}
function TranX( X : Reals) : Integer; { Transform X to Pixels }
begin
TranX:= Round((X - X0) * SlopeX + Xp0);
end; { TranX }
{--------------------------------------}
function TranY( Y : Reals) : Integer; { Transform Y to Pixels }
begin
TranY:= Round((Y - Y0) * SlopeY + Yp0);
end; { TranY }
{--------------------------------------}
procedure CompTran; { Compute constants for transforms to pixels }
{ Using point-slope form of linear transform }
begin
Ra:= 1.0;
SlopeX:= (1.0 + MaxXPix - 2.0 * Border) / (2.0 * Ra);
X0:= -Ra; Xp0:= Border;
SlopeY:= -SlopeX / AspectR;
Y0:= 0; Yp0:= (1.0 + MaxYPix) / 2.0;
if (TranY( Ra) * AspectR < Border) then begin
SlopeY:= -(1 + MaxYPix - 2.0 * Border / AspectR) / (2 * Ra);
Y0:= Ra; Yp0:= Border / AspectR;
SlopeX:= -SlopeY * AspectR;
X0:= 0; Xp0:= (1 + MaxXPix) / 2.0;
end;
end; { CompTran }
{--------------------------------------}
procedure Orbits; { Determine the orbits }
begin
for i:= 1 to n-1 do
A[i]:= i+1; { Set position i to point to next position }
A[n]:= 1;
p:= 1 + ((f mod n) + n - 2) mod n; { Init previous position based on f }
k:= 1; r:= n;
repeat
if (1 < k) and (k < n) then { If not first or last selection }
for i:= 1 to ((m-1) mod r) do { Move m-1 positions (zero OK) }
p:= A[p];
c:= A[p]; { Update current position }
B[k]:= c; c:= A[c]; { Select the object at c }
A[ A[p]]:= k; { Save execution order }
Inc( k); Dec( r);
A[p]:= c; { Re-link circular chain of pointers }
if KeyPressed then TestIt; { Abort run if Esc typed }
until k > n; { Until all are selected }
A[c]:= n; { A[c] was clobbered, so restore it }
s:= 0; { Count number of fixed elements }
for i:= 1 to n do
if B[i] = i then Inc( s);
end; { Orbits }
{--------------------------------------}
procedure ReportCyclic; { Generate cyclic notation output }
begin
Color:= ForceColor( Yellow);
MoveTo( 10, MaxYPix + 3 - 2 * TH3);
OutText('Permutation in cyclic notation:');
MoveTo( 10, MaxYPix + 3 - 1 * TH3);
p:= B[1]; c:= 1;
repeat
OutText('(');
repeat
if c <> 0 then begin
OutText( I2St( c));
A[p]:= 0; p:= c;
end;
c:= A[p];
if c <> 0 then OutText(', ');
until (c = 0) or (GetX > MaxXPix);
OutText(')');
i:= 0;
repeat
Inc( i); c:= A[i];
until (c <> 0) or (i = n);
if c<> 0 then p:= B[c];
until (c = 0) or (GetX > MaxXPix);
end; { ReportCyclic }
{--------------------------------------}
procedure PlotIt; { Plot the circle }
{
n
n-1 1
.
. n points equally spaced on the
. circumference of a circle
n/2
}
var
an : Reals; { Angle between points }
K1 : Reals; { Constant for 6 pixel tick mark }
K2 : Reals; { Constant for 15 pixel tick mark }
K3 : Reals; { Constant for 15 pixel + TextHeight tick mark }
Ss : String[1]; { String 's' or '' if s = 1 }
d : Integer; { Delta spacing for labeling points }
begin
if KeyPressed then Exit;
Ss:= 's';
if s = 1 then Ss:= '';
Ch:= ' ';
I:= 1;
ClearViewPort;
SetLineStyle(SolidLn, 0, NormWidth);
K1:= (SlopeY - 0.55 * TH3) / SlopeY;
K2:= (SlopeY - 1.36 * TH3) / SlopeY;
K3:= (SlopeY - 2.09 * TH3) / SlopeY;
an:= 2 * Pi / n; x:= 0;
Color:= ForceColor( Yellow);
MoveTo( 10, 1);
OutText('Graphical Orbits in Josephus permutations');
MoveTo( 10, TH3);
OutText('Type arrow keys to change m and f');
MoveTo( 10, 2 * TH3);
OutText('PgUp/PgDn to change n');
MoveTo( 10, 3 * TH3);
OutText('+/- for next');
MoveTo( 10, 4 * TH3);
OutText('Home to restart');
MoveTo( 10, 5 * TH3);
OutText('or Esc to quit');
MoveTo( 10, 6 * TH3);
OutText('n = ' + I2St(n) + ', m = ' + I2St(m) + ',');
MoveTo( 10, 7 * TH3);
OutText('f = ' + I2St(f));
MoveTo( 10, 8 * TH3);
OutText(I2St( s) + ' fixed point' + Ss);
MoveTo( 10, 9 * TH3);
OutText('# ' + I2St( B[n]) + ' last');
Color:= ForceColor( Green);
MoveTo( TranX( 0), TranY( 0));
LineTo( TranX( 0), TranY( 0));
Circle( TranX( 0), TranY(0), 10);
Circle( TranX( 0), TranY(0), Round( Ra * -SlopeY * AspectR));
x:= 1; d:= n div 50;
if d <= 1 then d:= 1
else if d <= 2 then d:= 2
else if d <= 5 then d:= 5
else if d <= 10 then d:= 10
else if d <= 20 then d:= 20
else if d <= 50 then d:= 50
else if d <= 100 then d:= 100
else if d <= 200 then d:= 200
else if d <= 500 then d:= 500;
repeat
X1:= Sin( x * an); Y1:= Cos( x * an);
if ((x mod d) = 0) or (x = n) then begin
X2:= K2 * X1; Y2:= K2 * Y1;
end
else begin
X2:= K1 * X1; Y2:= K1 * Y1;
end;
MoveTo( TranX( X1), TranY( Y1)); { Draw tick marks at each Point x }
LineTo( TranX( X2), TranY( Y2));
if (x mod d) = 0 then begin
X3:= K3 * X1; Y3:= K3 * Y1;
MoveTo( TranX( X3) - TextWidth(I2St( x)) div 2,
TranY( Y3) - TH div 2);
OutText( I2St( x));
end;
Inc( x);
until (x > n) or KeyPressed;
Color:= ForceColor( Yellow);
MoveTo( TranX( 0), TranY( Ra));
MoveRel( 2, - TH3 - 3);
OutText('n');
x:= 1; Color:= ForceColor( Yellow);
repeat
y:= A[x];
X1:= Sin( x * an); Y1:= Cos( x * an);
X3:= Sin( y * an); Y3:= Cos( y * an);
X2:= (X1 + 19 * X3) / 20; Y2:= (Y1 + 19 * Y3) / 20;
MoveTo( TranX( X1), TranY( Y1)); { Draw line from point x to Point y }
LineTo( TranX( X2), TranY( Y2));
Inc( x);
until (x > n) or KeyPressed;
SetLineStyle(SolidLn, 0, ThickWidth);
x:= 1; Color:= ForceColor( LightRed);
repeat
y:= A[x];
X1:= Sin( x * an); Y1:= Cos( x * an);
X3:= Sin( y * an); Y3:= Cos( y * an);
X2:= (X1 + 19 * X3) / 20; Y2:= (Y1 + 19 * Y3) / 20;
MoveTo( TranX( X2), TranY( Y2)); { Draw arrow head in red to Point y }
LineTo( TranX( X3), TranY( Y3));
Inc( x);
until (x > n) or KeyPressed;
Color:= ForceColor( White);
x:= 1;
repeat
y:= A[x];
if x = y then begin
X1:= Sin( x * an); Y1:= Cos( x * an);
MoveTo( TranX( X1), TranY( Y1)); { Draw black holes last }
LineTo( TranX( X1), TranY( Y1));
end;
Inc( x);
until (x > n) or KeyPressed;
ReportCyclic;
Ch:= ReadKey;
PreRead:= True;
end; { PlotIt }
{--------------------------------------}
procedure CompMaxm; { Compute Max m = (n-1)! or MaxInt }
begin
if n > 8 then Maxm:= MaxInt
else begin { Maxm = (n-1)! }
Maxm:= 1;
for i:= 2 to n-1 do Maxm:= i * Maxm;
end;
end; { CompMaxm }
{--------------------------------------}
begin { Joseph5 }
repeat
Init; { Initialize program }
CompTran; { Compute constants for transforms }
repeat
Orbits; { Determine the orbits }
PlotIt; { Plot the circle }
if not PreRead then Ch:= ReadKey; { Wait for operator action }
PreRead:= False;
if Ch = Chr(0) then Ch:= ReadKey;
CompMaxm;
if Ch = '+' then begin
Inc(f);
if f > n then begin
f:= 1; Inc(m);
if (m > Maxm) or (m < 0) then begin
m:= 1;
if n < Maxn then Inc(n)
else n:= 1;
end;
end;
end
else if Ch = '-' then begin
Dec(f);
if f < 1 then begin
f:= n; Dec(m);
if m < 1 then begin
if n > 1 then Dec(n);
f:= n;
CompMaxm;
m:= Maxm;
end;
end;
end
else begin
if Ch = Chr(73) then Inc(n); { Page Up }
if Ch = Chr(81) then Dec(n); { Page Down }
if Ch = Chr(72) then Inc(m); { Up arrow }
if Ch = Chr(80) then Dec(m); { Down arrow }
if Ch = Chr(75) then Dec(f); { Left arrow }
if Ch = Chr(77) then Inc(f); { Right arrow }
if n < 1 then n:= 1;
if n > Maxn then n:= Maxn;
if m < 0 then m:= 1; { Prevent m overflow }
if m = 0 then m:= MaxInt;
if m > MaxInt then m:= 1;
if f < 1 then f:= n;
if f > n then f:= 1;
end;
until (Ch = Chr(27)) or (Ch = Chr(71)); { Esc or Home }
RestoreCrtMode;
CloseGraph;
until Ch = Chr(27);
ExitProc; { Exit the program }
end. { Joseph5 }
{ End of file Joseph5.Pas **************************************************}
times since October 20, 2004.